교과서 질문
Use the graph of the greatest integer function y = ⌊x⌋, Figure 1.10 in Section 1.1, to help you find the limits in Exercises 21 and 22.
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b. limt→4−(t−⌊t⌋)
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Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
2장, 문제 2b
Finding Limits Graphically
Which of the following statements about the function y = f(x) graphed here are true, and which are false?
b. limx→2 f(x) does not exist
검증된 단계별 안내1
Examine the graph of the function y = f(x) around x = 2 to determine the behavior of the function as x approaches 2 from both the left and the right.
Identify the left-hand limit, which is the value that f(x) approaches as x approaches 2 from the left (x → 2⁻).
Identify the right-hand limit, which is the value that f(x) approaches as x approaches 2 from the right (x → 2⁺).
Compare the left-hand limit and the right-hand limit. For the overall limit lim(x→2) f(x) to exist, these two limits must be equal.
If the left-hand limit and the right-hand limit are not equal, then the limit lim(x→2) f(x) does not exist. Otherwise, if they are equal, the limit exists and is equal to that common value.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Limits
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps determine the value that a function approaches, even if it does not actually reach that value. Understanding limits is crucial for analyzing continuity, derivatives, and integrals.
추천 영상:
05:50
One-Sided Limits
Graphical Interpretation of Limits
Graphically, the limit of a function as x approaches a certain value can be observed by examining the behavior of the function's graph near that point. If the function approaches a specific y-value from both sides as x approaches a given value, the limit exists. However, if the function behaves differently from the left and right, the limit may not exist.
추천 영상:
6:47Finding Limits Numerically and Graphically
Existence of Limits
For a limit to exist at a point, the left-hand limit and right-hand limit must both exist and be equal. If there is a discontinuity, such as a jump or an asymptote at that point, the limit does not exist. This concept is essential for determining the validity of statements regarding limits in a given function.
추천 영상:
03:07Cases Where Limits Do Not Exist
관련 실천
교과서 질문
Finding One-Sided Limits Algebraically
Find the limits in Exercises 11–20.
a. limx→1+ (√2x (x − 1)) / |x − 1|
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교과서 질문
Average Rates of Change
In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.
g(x)=x²−2x
a. [1, 3]
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교과서 질문
Estimating Limits
[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.
Let g(θ) = (sinθ) / θ.
b. Support your conclusion in part (a) by graphing g near θ₀ = 0.
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교과서 질문
Suppose that limx→−2 p(x) = 4, limx→−2 r(x) = 0, and limx→−2 s(x) = −3. Find
a. limx→−2 (p(x) + r(x) + s(x))
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교과서 질문
Suppose limx→b f(x) = 7 and lim x→b g(x) = −3. Find
b. limx→b f(x)⋅g(x)
219
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